Goodie1857 Goodie1857

19-09-2019
Mathematics

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Use the binomial theorem to show that 7^n + 2 is divisible by 3, where n is a positive integer.
(Hint: Write 7 = 6 + 1.)

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LammettHash LammettHash

19-09-2019

By the binomial theorem,

[tex]7^n=(6+1)^n=6^n+n6^{n-1}+\dbinom n26^{n-2}+\cdots+\dbinom n{n-2}6^2+n6^1+1[/tex]

Each term containing a factor 6 is divisible by 3, since 3 divides 6. The remaining 1 and 2 sum to 3, which is also divisible by 3.

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